Antimicrobial combinations: Bliss independence and Loewe additivity derived from mechanistic multi-hit models

Antimicrobial peptides (AMPs) and antibiotics reduce the net growth rate of bacterial populations they target. It is relevant to understand if effects of multiple antimicrobials are synergistic or antagonistic, in particular for AMP responses, because naturally occurring responses involve multiple AMPs. There are several competing proposals describing how multiple types of antimicrobials add up when applied in combination, such as Loewe additivity or Bliss independence. These additivity terms are defined ad hoc from abstract principles explaining the supposed interaction between the antimicrobials. Here, we link these ad hoc combination terms to a mathematical model that represents the dynamics of antimicrobial molecules hitting targets on bacterial cells. In this multi-hit model, bacteria are killed when a certain number of targets are hit by antimicrobials. Using this bottom-up approach reveals that Bliss independence should be the model of choice if no interaction between antimicrobial molecules is expected. Loewe additivity, on the other hand, describes scenarios in which antimicrobials affect the same components of the cell, i.e. are not acting independently. While our approach idealizes the dynamics of antimicrobials, it provides a conceptual underpinning of the additivity terms. The choice of the additivity term is essential to determine synergy or antagonism of antimicrobials.This article is part of the themed issue 'Evolutionary ecology of arthropod antimicrobial peptides'.